My code (arr is array of native BigInt):

  console.info("My number as BN:     " + BN(arr[0] * arr[0] + arr[1] * arr[1]).toFixed());
  console.info("My number as BigInt: " + (arr[0] * arr[0] + arr[1] * arr[1]));

Output:

Testing  [
  10, 7,  5,  5, 3, 2,  7, 6, 5,  9,
  10, 2,  6,  7, 2, 9, 10, 8, 8,  1,
  10, 8,  8,  6, 5, 6,  4, 5, 9,  9,
   4, 8, 10, 10, 4, 5,  7, 6, 6, 10
]
My number as BN:     352198124718383079462822912000000000000
My number as BigInt: 352198124718383079462822912000000000000
Actual          6211882367758080000,17709055371970560000
Possible expect 15840161589507840000,10064164423173120000
Product to get 3.52198124718383079462822912e+38
Your A*A + B*B 3.52198124718383079462822912e+38
Correct

As you see, BN.toFixed is able to get accurate representation.

Why not print BigInts as strings, then? Is it difficult to convert bigint to string and display it?

Not easy (because Bignumber.js doesnt't display well very big numbers) but I'll try to explain that your result is wrong.
with JS

you: A = 3; B = 38575890503412110
me : A = 20534466401280000; B = 32656316659200000
your product: 1.488099328131240613479552494652109e+33
my   product: 1.4880993281312406429302784e+33

They seem pretty close.
with Python or another language with native big integers:

you: A = 3; B = 38575890503412110
me : A = 20534466401280000; B = 32656316659200000
your product: 1,488,099,328,131,240,613,479,552,494,652,109
my   product: 1,488,099,328,131,240,642,930,278,400,000,000
mine - your =                        29,450,725,905,347,891 ~= 2.9450725905347892e+16

I expect my answer is clear enough, if not please tell.

I looked at your code and trie it and unfortunately I am sorry to think that you are not in the right way.

Edit: It seems you now Python so you can try it on this kata and have no problem with big integeers.